In deriving the Feynman diagrams of Figure REF  we consider the connected Green's function from the
beginning [1]}, since the disconnected diagrams are canceled by the denominator function and
apply the \({\mathbb {Z}}_-\)  operator,
\({\mathbb {Z}}_-\left[\exp \left[-i S_{\rm int}(\delta J_{\pm })\right]\exp \left[-i S_{\rm free}(J_\pm )\right]\right]_{\rm conn.}\) 
